Skip to content

Latest commit

 

History

History
49 lines (36 loc) · 2.24 KB

objective_function.md

File metadata and controls

49 lines (36 loc) · 2.24 KB

The objective function, combinatorial optimization problems in graph theory. This kind of problem can be translated into a Quadratic Unconstrained Binary Optimization (QUBO) problem, suitable for solving on a D-Wave quantum annealer.

Mathematical Explanation:

  • ( \sum_{i \in V} x_i ): Summation of the variables ( x_i ) over all vertices ( i ) in ( V ).
  • ( \gamma \sum_{(i, j) \in E}(1-x_i-x_j+x_i \cdot x_j) ): A weighted sum over all edges ( (i, j) ) in ( E ). The ( \gamma ) term scales the sum.

The function seems to minimize both the individual ( x_i ) variables and their relationships across edges, scaled by a factor ( \gamma ).

Python Code for QUBO Problem with D-Wave

To solve this problem using a D-Wave quantum computer, first translate it into a QUBO problem. Here's a Python snippet using D-Wave's Ocean SDK to set up this problem:

from dwave.system import LeapHybridSampler
from dimod import BinaryQuadraticModel
import networkx as nx

# Create a simple graph
G = nx.Graph()
G.add_edges_from([(0, 1), (1, 2), (2, 3), (3, 0)])

# Initialize variables
gamma = 0.5  # Example gamma value

# Create an empty Binary Quadratic Model
bqm = BinaryQuadraticModel('BINARY')

# Add the linear terms to the BQM (corresponding to the first term in your equation)
for i in G.nodes:
    bqm.add_variable(i, 1)  # Coefficient is 1 for x_i

# Add the quadratic terms to the BQM (corresponding to the second term in your equation)
for i, j in G.edges:
    bqm.add_interaction(i, j, -2 * gamma)  # Coefficient for x_i * x_j is -2*gamma
    bqm.add_offset(gamma)  # Constant term gamma

# Use the LeapHybridSampler to find the solution
sampler = LeapHybridSampler()
sampleset = sampler.sample(bqm)

# Print the results
print(sampleset)

In this Python3 code, we've set up the QUBO problem that corresponds to the optimization problem provided. Note that D-Wave's quantum annealers work best with problems that can be cast as Ising models or QUBO problems, both of which involve quadratic terms—perfect for the problem at hand.